Problem: $f(x) = \begin{cases} -4 & \text{if } x = 2 \\ -2x^{2}+2 & \text{otherwise} \end{cases}$ What is the range of $f(x)$ ?
Solution: First consider the behavior for $x \ne 2$ Consider the range of $-2x^{2}$ The range of $x^2$ is $\{\, y \mid y \ge 0 \,\}$ Multiplying by $-2$ flips the range to $\{\, y \mid y \le 0 \,\}$ To get $-2x^{2}+2$ , we add $2$ If $x = 2$, then $f(x) = -4$. Since $-4 ≤ 2$, the range is still $\{\, y \mid y ≤ 2 \,\}$.